Created
May 31, 2017 14:10
-
-
Save ashiato45/0d0d0272dc6b2dc0ab22db2673a35d6b to your computer and use it in GitHub Desktop.
This file contains bidirectional Unicode text that may be interpreted or compiled differently than what appears below. To review, open the file in an editor that reveals hidden Unicode characters.
Learn more about bidirectional Unicode characters
sig Element {} | |
one sig Ring{ | |
elements: set Element, | |
zero: one elements, | |
un: one elements, | |
add: elements -> elements -> one elements, | |
mult: elements -> elements -> one elements, | |
neg: elements -> one elements | |
} | |
fact NoRedundantElements{ | |
all e: Element | e in Ring.elements | |
} | |
fact AddZero{ | |
all a: Ring.elements | Ring.add [a] [Ring.zero] = a | |
all a: Ring.elements | Ring.add [Ring.zero] [a] = a | |
} | |
fact AddAssociative{ | |
all a, b, c: Ring.elements | Ring.add [a] [Ring.add [b] [c]] = Ring.add [Ring.add [a] [b]] [c] | |
} | |
fact AddCommutative{ | |
all a, b: Ring.elements | Ring.add [a] [b] = Ring.add [b] [a] | |
} | |
fact Neg{ | |
all a: Ring.elements | Ring.add [a] [Ring.neg [a]] = Ring.zero | |
} | |
fact MultAssociative{ | |
all a, b, c: Ring.elements | Ring.mult [a] [Ring.mult [b] [c]] = Ring.mult [Ring.mult [a] [b]] [c] | |
} | |
// Commutative Ring! | |
fact MultCommutative{ | |
all a, b: Ring.elements | Ring.mult [a] [b] = Ring.mult [b] [a] | |
} | |
fact MultDistributive{ | |
all a, b, c: Ring.elements | Ring.mult [a] [Ring.add [b] [c]] = Ring.add [Ring.mult [a] [b]] [Ring.mult [a] [c]] | |
} | |
fact MultUn{ | |
all a: Ring.elements | Ring.mult [a] [Ring.un] = a | |
} | |
// run {} | |
pred isIdeal(i: set Element){ | |
Ring.zero in i | |
all a, b: i | Ring.add [a] [b] in i | |
all a:i , c: Ring.elements | Ring.mult [c] [a] in i | |
} | |
//run {some i: set Element | isIdeal [i]} | |
assert main{ | |
all i: set Element, a: set Element, b: set Element | | |
((i in a+b) and | |
isIdeal[i] and | |
isIdeal[a] and | |
isIdeal[b]) implies | |
((i in a) or (i in b)) | |
} | |
//check main for 10 //valid | |
assert main2{ | |
all i: set Element, a: set Element, b: set Element, c: set Element | | |
((i in a+b+c) and | |
isIdeal[i] and | |
isIdeal[a] and | |
isIdeal[b] and | |
isIdeal[c]) implies | |
((i in a) or (i in b) or (i in c)) | |
} | |
//check main2 for 8 | |
pred isMultClosed(i: set Element){ | |
Ring.un in i | |
all a, b: i | Ring.mult [a] [b] in i | |
} | |
//run {some i: set Element | isMultClosed [i] } | |
pred pEquiv(a, s, b, t: Element){ //pseudo equivalent | |
Ring.mult [a] [t] = Ring.mult [b] [s] | |
} | |
/* run {some m: set Element, a, b, s, t: Element | | |
s in m and | |
t in m and | |
pEquiv [a] [s] [b] [t] | |
} */ | |
assert main3{ | |
all mc: set Element, a, b, c, s, t, u: Element | | |
(isMultClosed [mc] and | |
s in mc and | |
t in mc and | |
u in mc and | |
pEquiv [a] [s] [b] [t] and | |
pEquiv [b] [t] [c] [u]) implies | |
pEquiv [a] [s] [c] [u] | |
} | |
// check main3 | |
assert main4{ | |
all mc: set Element, a, b, c, s, t, u: Element | | |
(isMultClosed [mc] and | |
s in mc and | |
t in mc and | |
u in mc and | |
pEquiv [a] [s] [b] [t] and | |
pEquiv [b] [t] [c] [u] and | |
not Ring.zero in mc) implies | |
pEquiv [a] [s] [c] [u] | |
} | |
check main4 for 4 |
Sign up for free
to join this conversation on GitHub.
Already have an account?
Sign in to comment